Mathematics

Problem

To prove the identity

\dfrac{\sin\theta}{1+\cos\theta}=\dfrac{1-\cos\theta}{\sin\theta}

, a student multiplies the numerator and denominator of the left-hand side by

(1-\cos\theta)

and gets

\dfrac{\sin\theta(1-\cos\theta)}{1-\cos^2\theta}

. Which of the following next steps will lead directly to the right-hand side? a) Factor the denominator as

(1-\cos\theta)(1+\cos\theta)

and cancel the common

(1-\cos\theta)

term b) Replace

1-\cos^{2}\theta

with

\tan^{2}\theta

and cancel factors c) Replace

1-\cos^{2}\theta

with

\sin^{2}\theta

and cancel one

\sin\theta

d) Replace

1-\cos^{2}\theta

with

\sin^{2}\theta

and rewrite the numerator as

(1-\cos\theta)^{2}